Abstract
Harmonic amoebas are generalisations of amoebas of algebraic curves immersed in complex tori. Introduced by Krichever in 2014, the consideration of such objects suggests to enlarge the scope of tropical geometry. In the present paper, we introduce the notion of harmonic morphisms from tropical curves to affine spaces and show how these morphisms can be systematically described as limits of families of harmonic amoeba maps on Riemann surfaces. It extends previous results about approximation of tropical curves in affine spaces and provides a different point of view on Mikhalkin’s approximation Theorem for regular phase-tropical morphisms, as stated e.g. by Mikhalkin in 2006. The results presented here follow from the study of imaginary normalised differentials on families of punctured Riemann surfaces and suggest interesting connections with compactifications of moduli spaces.
Highlights
The present work is an attempt to give an appropriate definition of tropical convergence for families of abstract algebraic curves, with a view towards the constructive and enumerative aspects of tropical geometry
We aim to understand the relations between the moduli of a family of algebraic curves converging in the sense of (1) and the limiting tropical curve
Theorem 1 will arise as a consequence of Theorem 3 describing the limit of imaginary normalised differentials on families of algebraic curves converging in the sense of Definition 1.1, see Sect. 3.1
Summary
The present work is an attempt to give an appropriate definition of tropical convergence for families of abstract algebraic curves, with a view towards the constructive and enumerative aspects of tropical geometry. The link between algebraic and tropical geometry is given by the notion of amoeba introduced in [10]. A family of algebraic curves {Ct }t>1 ⊂ (C )m is said to converge to a tropical curve C ⊂ Rm if lim t →∞. We aim to understand the relations between the moduli of a family of algebraic curves converging in the sense of (1) and the limiting tropical curve. We want to understand how to prescribe the moduli of a family of algebraic curves so that the latter family converges in the sense of (1). It will be convenient to leave the algebraic setting and consider the more general theory of harmonic amoebas introduced in [12]. We will come back to the algebraic setting and reprove Mikhalkin’s approximation Theorem
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